<h1>Results</h1>


The program performs two tasks, the first being to generate a
visualization of the mapped domain, the second to compute pi by the
two methods described. Let us first take a look at the generated
graphics. They are generated in Gnuplot format, and can be viewed with
the commands
@code
set style data lines
set size ratio -1
unset key
unset xtics
unset ytics
plot [-1:1][-1:1] "ball_0_mapping_q_1.dat" lw 4 lt rgb "black"
@endcode
or using one of the other filenames. The second line makes sure that
the aspect ratio of the generated output is actually 1:1, i.e. a
circle is drawn as a circle on your screen, rather than as an
ellipse. The third line switches off the key in the graphic, as that
will only print information (the filename) which is not that important
right now. Similarly, the fourth and fifth disable tick marks. The plot
is then generated with a specific line width ("lw", here set to 4)
and line type ("lt", here chosen by saying that the line should be
drawn using the RGB color "black").

The following table shows the triangulated computational domain for $Q_1$,
$Q_2$, and $Q_3$ mappings, for the original coarse grid (left), and a once
uniformly refined grid (right).

<div class="twocolumn" style="width: 80%">
  <div>
    <img src="https://www.dealii.org/images/steps/developer/step_10_ball_0_q1.svg"
         alt="Five-cell discretization of the disk."
         width="400" height="400">
  </div>
  <div>
    <img src="https://www.dealii.org/images/steps/developer/step_10_ball_1_q1.svg"
         alt="20-cell discretization of the disk (i.e., five cells
              refined once)."
         width="400" height="400">
  </div>
  <div>
    <img src="https://www.dealii.org/images/steps/developer/step_10_ball_0_q2.svg"
         alt="Five-cell discretization of the disk with quadratic edges. The
              boundary is nearly indistinguishable from the actual circle."
         width="400" height="400"
         >
  </div>
  <div>
    <img src="https://www.dealii.org/images/steps/developer/step_10_ball_1_q2.svg"
         alt="20-cell discretization with quadratic edges."
         width="400" height="400">
  </div>
  <div>
    <img src="https://www.dealii.org/images/steps/developer/step_10_ball_0_q3.svg"
         alt="Five-cell discretization of the disk with cubic edges. The
              boundary is nearly indistinguishable from the actual circle."
         width="400" height="400">
  </div>
  <div>
    <img src="https://www.dealii.org/images/steps/developer/step_10_ball_1_q3.svg"
         alt="20-cell discretization with cubic edges."
         width="400" height="400">
  </div>
</div>

These pictures show the obvious advantage of higher order mappings: they
approximate the true boundary quite well also on rather coarse meshes. To
demonstrate this a little further, here is part of the upper right quarter
circle of the coarse meshes with $Q_2$ and $Q_3$ mappings, where the dashed
red line marks the actual circle:

<div class="twocolumn" style="width: 80%">
  <div>
    <img src="https://www.dealii.org/images/steps/developer/step_10_exact_vs_interpolate_q2.svg"
         alt="Close-up of quadratic discretization. The distance between the
         quadratic interpolant and the actual circle is small."
         width="400" height="400">
  </div>
  <div>
    <img src="https://www.dealii.org/images/steps/developer/step_10_exact_vs_interpolate_q3.svg"
         alt="Close-up of cubic discretization. The distance between the
         cubic interpolant and the actual circle is very small."
         width="400" height="400">
  </div>
</div>

Obviously the quadratic mapping approximates the boundary quite well,
while for the cubic mapping the difference between approximated domain
and true one is hardly visible already for the coarse grid. You can
also see that the mapping only changes something at the outer
boundaries of the triangulation. In the interior, all lines are still
represented by linear functions, resulting in additional computations
only on cells at the boundary. Higher order mappings are therefore
usually not noticeably slower than lower order ones, because the
additional computations are only performed on a small subset of all
cells.



The second purpose of the program was to compute the value of pi to
good accuracy. This is the output of this part of the program the last
time it was updated:
@code
Output of grids into gnuplot files:
===================================
Refinement level: 0
Degree = 1
Degree = 2
Degree = 3

Refinement level: 1
Degree = 1
Degree = 2
Degree = 3

Computation of Pi by the area:
==============================
Degree = 1
cells      eval.pi            error
    5 1.9999999999999984 1.1416e+00    -
   20 2.8284271247461947 3.1317e-01 1.87
   80 3.0614674589207289 8.0125e-02 1.97
  320 3.1214451522579965 2.0148e-02 1.99
 1280 3.1365484905461294 5.0442e-03 2.00
 5120 3.1403311569547410 1.2615e-03 2.00

Degree = 2
cells      eval.pi            error
    5 3.1045694996615865 3.7023e-02    -
   20 3.1391475703122351 2.4451e-03 3.92
   80 3.1414377167038401 1.5494e-04 3.98
  320 3.1415829366418513 9.7169e-06 4.00
 1280 3.1415920457578785 6.0783e-07 4.00
 5120 3.1415926155920988 3.7998e-08 4.00

Degree = 3
cells      eval.pi            error
    5 3.1410033851499315 5.8927e-04    -
   20 3.1415830393583946 9.6142e-06 5.94
   80 3.1415925017363939 1.5185e-07 5.98
  320 3.1415926512106185 2.3792e-09 6.00
 1280 3.1415926535527783 3.7015e-11 6.01
 5120 3.1415926535891936 5.9952e-13 5.95

Degree = 4
cells      eval.pi            error
    5 3.1415871927401144 5.4608e-06    -
   20 3.1415926314742491 2.2116e-08 7.95
   80 3.1415926535026268 8.7166e-11 7.99
  320 3.1415926535894005 3.9257e-13 7.79
 1280 3.1415926535899774 1.8430e-13 1.09
 5120 3.1415926535897669 2.6201e-14 2.81

Computation of Pi by the perimeter:
===================================
Degree = 1
cells      eval.pi            error
    5 2.8284271247461903 3.1317e-01    -
   20 3.0614674589207187 8.0125e-02 1.97
   80 3.1214451522580493 2.0148e-02 1.99
  320 3.1365484905459371 5.0442e-03 2.00
 1280 3.1403311569547530 1.2615e-03 2.00
 5120 3.1412772509327755 3.1540e-04 2.00

Degree = 2
cells      eval.pi            error
    5 3.1248930668550590 1.6700e-02    -
   20 3.1404050605605454 1.1876e-03 3.81
   80 3.1415157631807005 7.6890e-05 3.95
  320 3.1415878042798600 4.8493e-06 3.99
 1280 3.1415923498174543 3.0377e-07 4.00
 5120 3.1415926345931982 1.8997e-08 4.00

Degree = 3
cells      eval.pi            error
    5 3.1414940401456053 9.8613e-05    -
   20 3.1415913432549156 1.3103e-06 6.23
   80 3.1415926341726905 1.9417e-08 6.08
  320 3.1415926532906924 2.9910e-10 6.02
 1280 3.1415926535851346 4.6585e-12 6.00
 5120 3.1415926535897158 7.7272e-14 5.91

Degree = 4
cells      eval.pi            error
    5 3.1415921029432572 5.5065e-07    -
   20 3.1415926513737582 2.2160e-09 7.96
   80 3.1415926535810699 8.7232e-12 7.99
  320 3.1415926535897576 3.5527e-14 7.94
 1280 3.1415926535897896 3.5527e-15 3.32
 5120 3.1415926535897940 8.8818e-16 2.00
@endcode

@note Once the error reaches a level on the
  order of $10^{-13}$ to $10^{-15}$, it is essentially dominated by
  round-off and consequently dominated by what precisely the library is doing
  in internal computations. Since these things change, the precise values
  and errors change from release to release at these round-off levels,
  though the overall order of errors should of course remain the same.
  See also the comment below in the section on
  <a href="#extensions">Possibilities for extensions</a> about how to compute
  these results more accurately.

One of the immediate observations from the output above is that in all cases
the values converge quickly to the true value of
$\pi=3.141592653589793238462643$. Note that for the $Q_4$ mapping, we are
already in the regime of roundoff errors and the convergence rate levels off,
which is already quite a lot. However, also note that for the $Q_1$ mapping,
even on the finest grid the accuracy is significantly worse than on the coarse
grid for a $Q_3$ mapping!



The last column of the output shows the convergence order, in powers of the
mesh width $h$. In the introduction, we had stated that the convergence order
for a $Q_p$ mapping should be $h^{p+1}$. However, in the example shown, the
order is rather $h^{2p}$! This at first surprising fact is explained by the
properties of the $Q_p$ mapping. At order <i>p</i>, it uses support points
that are based on the <i>p</i>+1 point Gauss-Lobatto quadrature rule that
selects the support points in such a way that the quadrature rule converges at
order 2<i>p</i>. Even though these points are here only used for interpolation
of a <i>p</i>th order polynomial, we get a superconvergence effect when
numerically evaluating the integral, resulting in the observed high order of
convergence. (This effect is also discussed in detail in the following
publication: A. Bonito, A. Demlow, and J. Owen: "A priori error
estimates for finite element approximations to eigenvalues and
eigenfunctions of the Laplace-Beltrami operator", submitted, 2018.)



<h3>Possibilities for extensions</h3>
<a name="extensions"></a>

As the table of numbers copied from the output of the program shows above,
it is not very difficult to compute the value of $\pi$ to 13 or 15 digits. But,
the output also shows that once we approach the level of accuracy with which
`double` precision numbers store information (namely, with roughly 16 digits
of accuracy), we no longer see the expected convergence order and the error
no longer decreases with mesh refinement as anticipated. This is because both
within this code and within the many computations that happen within deal.II
itself, each operation incurs an error on the order of $10^{-16}$; adding
such errors many times over then results in an error that may be on the
order of $10^{-14}$, which will dominate the discretization error after
a number of refinement steps and consequently destroy the convergence rate.

The question is whether one can do anything about this. One thought is to
use a higher-precision data type. For example, one could think of declaring
both the `area` and `perimeter` variables in `compute_pi_by_area()` and
`compute_pi_by_perimeter()` with data type `long double`. `long double`
is a data type that is not well specified in the C++ standard but at least
on Intel processors has around 19, instead of around 16, digits of accuracy.
If we were to do that, we would get results that differ from the ones shown
above. However, maybe counter-intuitively, they are not uniformly better.
For example, when computing $\pi$ by the area, at the time of writing
these sentences we get these values with `double` precision for degree 4:
@code
    5 3.1415871927401144 5.4608e-06    -
   20 3.1415926314742491 2.2116e-08 7.95
   80 3.1415926535026268 8.7166e-11 7.99
  320 3.1415926535894005 3.9257e-13 7.79
 1280 3.1415926535899774 1.8430e-13 1.09
 5120 3.1415926535897669 2.6201e-14 2.81
@endcode
On the other hand, the results are as follows when using `long double`:
@code
 cells      eval.pi            error
    5 3.1415871927401136 5.4608e-06    -
   20 3.1415926314742446 2.2116e-08 7.95
   80 3.1415926535026215 8.7172e-11 7.99
  320 3.1415926535894516 3.4157e-13 8.00
 1280 3.1415926535897918 1.5339e-15 7.80
 5120 3.1415926535897927 5.2649e-16 1.54
@endcode
Indeed, here we get results that are approximately 50 times as accurate.
On the other hand, when computing $\pi$ by the perimeter, we get this with
`double` precision:
@code
    5 3.1415921029432572 5.5065e-07    -
   20 3.1415926513737582 2.2160e-09 7.96
   80 3.1415926535810699 8.7232e-12 7.99
  320 3.1415926535897576 3.5527e-14 7.94
 1280 3.1415926535897896 3.5527e-15 3.32
 5120 3.1415926535897940 8.8818e-16 2.00
@endcode
Whereas we get the following with `long double`:
@code
    5 3.1415921029432572 5.5065e-07     -
   20 3.1415926513737595 2.2160e-09  7.96
   80 3.1415926535810703 8.7230e-12  7.99
  320 3.1415926535897576 3.5705e-14  7.93
 1280 3.1415926535897918 1.3785e-15  4.70
 5120 3.1415926535897944 1.3798e-15 -0.00
@endcode
Here, using `double` precision is more accurate by about a factor of
two. (Of course, in all cases, we have computed $\pi$ with more
accuracy than any engineer would ever want to know.)

What explains this unpredictability? In general, round-off errors can
be thought of as random, and add up in ways that are not worth thinking
too much about; we should therefore always treat any accuracy beyond, say,
thirteen digits as suspect. Thus, it is probably not worth spending
too much time on wondering why we get different winners and losers in the
data type exchange from `double` and `long double`. The accuracy of the
results is also largely not determined by the precision of the data type
in which we accumulate each cell's (or face's) contributions, but the
accuracy of what deal.II gives us via FEValues::JxW() and FEFaceValues::JxW(),
which always uses `double` precision and which we cannot directly affect.

But there are cases where one can do something about the precision, and it
is worth at least mentioning the name of the most well-known algorithm in
this area. Specifically, what we are doing when we add contributions into
the `area` and `perimeter` values is that we are adding together *positive*
numbers as we do here. In general, the round-off errors associated with each
of these numbers is random, and if we add up contributions of substantially
different sizes, then we will likely be dominated by the error in the largest
contributions. One can avoid this by adding up numbers sorted by their
size, and this may then result in marginally more accurate end results.
The algorithm that implements this is typically called
<a href="https://en.wikipedia.org/wiki/Kahan_summation_algorithm">Kahan's summation algorithm</a>.
While one could play with it in the current context, it is likely not going
to improve the accuracy in ways that will truly matter.
